The structural (nonlinear) stability of a shock wave means the local-in-time existence and uniqueness of a smooth solution \((U^+,U^-,\varphi )\)  to problem (REF )–(REF ). It is well-known that the Lax's \(k\) –shock conditions [1]}, which guarantee the correct number of boundary conditions, are necessary for the well-posedness of such free boundary problem. It was shown in [2]} that, as in gas dynamics, only extreme shocks are possible in elastodynamics. If, without loss of generality, we assume that \(\mathfrak {m}>0\) , then they are 1-shocks. The 1-shock conditions for (REF ) read [2]}:
\(\left\lbrace \begin{array}{l}v_{\rm N}^--\partial _t\varphi >\sqrt{(\bar{c}^-)^2+(F_{1{\rm N}}^{-})^2+(F_{2{\rm N}}^{-})^2}\,, \\[6pt]\sqrt{(F_{1{\rm N}}^{+})^2+(F_{2{\rm N}}^{+})^2}<v_N^+-\partial _t\varphi <\sqrt{(\bar{c}^{+})^2+(F_{1{\rm N}}^{+})^2+(F_{2{\rm N}}^{+})^2} \quad \mbox{on}\ \Gamma (t),\end{array}\right.\) 
